Proposition 1.

For the proof, we use the following setup (see the parent entry for more detail):

E=(S,Σ,δ,I,F,ϵ)ESnormal-ΣδIFϵE=(S,\Sigma,\delta,I,F,\epsilon) is an ϵϵ\epsilon-automaton, and EϵsubscriptEϵE_{{\epsilon}} is the automaton associated with EEE,

h:(Σ∪{ϵ})*→Σ*normal-:hnormal-→superscriptnormal-Σϵsuperscriptnormal-Σh:(\Sigma\cup\{\epsilon\})^{*}\to\Sigma^{*} is the homomorphism that erases ϵϵ\epsilon (it takes ϵϵ\epsilon to the empty word, also denoted by ϵϵ\epsilon). From the parent entry, L⁢(E):=h⁢(L⁢(Eϵ))assignLEhLsubscriptEϵL(E):=h(L(E_{{\epsilon}})).

Proof.

Define a functionδ1:S×Σ→P⁢(S)normal-:subscriptδ1normal-→Snormal-ΣPS\delta_{1}:S\times\Sigma\to P(S), as follows: for each pair (s,a)∈S×ΣsaSnormal-Σ(s,a)\in S\times\Sigma, let

In other words, δ1⁢(s,a)subscriptδ1sa\delta_{1}(s,a) is the set of all statesreachable from sss by words of the form ϵm⁢a⁢ϵnsuperscriptϵmasuperscriptϵn\epsilon^{m}a\epsilon^{n}. As usual, we extend δ1subscriptδ1\delta_{1} so its domain is S×Σ*Ssuperscriptnormal-ΣS\times\Sigma^{*}. By abuse of notation, we use δ1subscriptδ1\delta_{1} again for this extension. First, we set δ1⁢(s,ϵ):={s}assignsubscriptδ1sϵs\delta_{1}(s,\epsilon):=\{s\}. Then we inductively define δ1⁢(s,u⁢a)=δ1⁢(δ1⁢(s,u),a)subscriptδ1suasubscriptδ1subscriptδ1sua\delta_{1}(s,ua)=\delta_{1}(\delta_{1}(s,u),a). Using induction,

In other words, if u=a1⁢a2⁢⋯⁢anusubscripta1subscripta2normal-⋯subscriptanu=a_{1}a_{2}\cdots a_{n}, then δ1⁢(s,u)subscriptδ1su\delta_{1}(s,u) is the set of all states reachable from sss by words of the form